As shown in the figure below, a uniform beam is supported by...

As shown in the figure below, a uniform beam is supported by a cable at one end and the force of friction at the other end. The cable makes an angle of ? = 30°, the length of the beam is L = 3.50 m, the coefficient of static friction between the wall and the beam is ?s = 0.400, and the weight of the beam is represented by w. Determine the minimum distance x from point A at which an additional weight 2w (twice the weight of the rod) can be hung without causing the rod to slip at point A. x = Since this is an equilibrium problem, we expect to use the first and second conditions of equilibrium to solve the problem. Before starting the problem, a well-drawn figure with all forces and all forces resolved into components would be a real asset to finding a solution. See if you can use the first condition of equilibrium to obtain an expression for the normal force and then the force of friction. Next see if you can use the first condition of equilibrium to obtain an expression for the tension in the cable. Finally see if you can use the second condition of equilibrium to obtain an expression that will allow you to determine the maximum distance you can hang the weight from the end of the beam supported by the wall. m

Transcript

00:01 So in this question, which is copied down here, you are giving your uniform beam, right? which is here.

00:08 So this is a beam, right? i'm just better.

00:13 So this is the beam here.

00:16 That's the beam.

00:18 And this team is in equilibrium by two forces.

00:24 One is of course this strain, which is pouring this beam on one end.

00:30 And another is due to the three.

00:32 The friction are with the war, right? the friction is going upward because the beam tends to go down by gravity, right? and therefore, the friction has to go upward so that the beam can stay in equilibrium.

00:51 So i think the gravity of the beam, since the be uniform, the gravity acts at the center of the beam, right? it's something going like this going down.

00:59 I'll write the friction to be a if, little f and the force by the strain is t, right? and this graph is g.

01:09 And of course, there will be a normal force acting on this beam by the world.

01:19 And similarly, the beam is pushing the world by the same amount of force.

01:24 I call this force n, right? so on the beam, there are altogether four forces, actually not just for force.

01:30 Actually, it's not force here.

01:31 I forgot.

01:32 There's a force which is actually here due to the weight at this point, right? so that's another force.

01:42 So i call this first, let's say, i just call w, right? so this w is just the mass of this block, which is two, oh, it's already given, it's two w, actually.

01:53 So i just called two w instead of given any additional semblance.

01:57 So the force here is just two w, right? so actually on this beam acting four forces, okay, five forces, sorry, five forces.

02:06 And of course, all of these forces must be balanced.

02:10 And also their torques must be balanced as well, so that the beam can stay in equilibrium.

02:15 And you can imagine that, then the question we ask you, how close can this weight? how close can this guy, can this weight be held to the world? so that the beams steers this equilibrium, right? so you need to find the critical value of x, which is the distance of this weight from the war, right? so let's write a set of equations that gives you relation between the forces.

02:52 The first equation i would like to write down is just the balance of forces.

03:00 That is the forces acting vertically must all be balanced, right? so there are a few forces acting upwards.

03:09 So first you have this first, right, which has a upward, has a vertical component, which is going upward, and it has a component going this way as well.

03:20 And that upward component has a magnitude to which t times santa, right? as one of the upward force.

03:31 Another upward force, of course, is the a for the friction, which is going upward as well.

03:37 And so these are two upward force, and this must be balanced by the two downward force.

03:44 One is the two w, right, due to the weight, and the other is the g, right, of the gravity of the beam itself.

03:53 So this is equation one.

03:56 And of course, we can write down a few other equations.

04:01 For example, for the horizontal force, it has to balance as well.

04:04 For the horizontal force, there are only two of them.

04:07 One is the horizontal component of the t, which is going to the left, and the other one is the normal force, n going to wrap.

04:15 These two balance each other...

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